L(s) = 1 | + (−2.52 − 4.54i)3-s + 8.01i·5-s − 12.6i·7-s + (−14.2 + 22.9i)9-s + 37.7·11-s − 60.0·13-s + (36.3 − 20.2i)15-s − 37.0i·17-s + 127. i·19-s + (−57.2 + 31.7i)21-s − 56.9·23-s + 60.8·25-s + (140. + 7.09i)27-s + 220. i·29-s + 2.26i·31-s + ⋯ |
L(s) = 1 | + (−0.485 − 0.874i)3-s + 0.716i·5-s − 0.680i·7-s + (−0.528 + 0.848i)9-s + 1.03·11-s − 1.28·13-s + (0.626 − 0.347i)15-s − 0.528i·17-s + 1.54i·19-s + (−0.594 + 0.330i)21-s − 0.516·23-s + 0.486·25-s + (0.998 + 0.0505i)27-s + 1.41i·29-s + 0.0130i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.292922121\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.292922121\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.52 + 4.54i)T \) |
good | 5 | \( 1 - 8.01iT - 125T^{2} \) |
| 7 | \( 1 + 12.6iT - 343T^{2} \) |
| 11 | \( 1 - 37.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 37.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 127. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 56.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 220. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.26iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 166.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 154. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 53.4iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 591.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 538. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 586.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 431.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 175. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 29.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 937.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 409. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 921. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.64e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.95303835323858351884885626679, −10.32828563772682551352308816672, −9.203859212256460262280643232213, −7.82386254415382805496995250845, −7.15988923245404222872060612193, −6.45124872015600752368310275803, −5.30885035340003946996814846208, −3.92883475140414740827658904723, −2.48561398859408122833865195196, −1.08529974382968151725005444470,
0.56200649776354457573490389071, 2.50159993461140881264264914740, 4.09251670012303824890582979926, 4.87565269675883492328852171934, 5.82697931887259892898983205000, 6.87736260160294195281294721021, 8.352593337217816125729359018914, 9.242577755015059649977664580524, 9.692065184334147082520020252357, 10.89208422341342022447517758177